Workflow

All the reported experiments were performed using Epimod, a tool recently developed by our group to provide a generalframework to draw and analyze epidemiological systems. For more details regarding the workflow of analysis see SIR, where it is exploited the SIR model as a simple step by step guide of the package.

Repository structure

Folders:

  1. Main: here the two main R files are stored: 1) run.R where it is showed how to start, to calibrate the model and to run the main analysis; 2) runDifferentScenario.R where there are the 16 function calls for simulating the 16 scenarios pictured in Figures 5 and 6.
  2. net: the PNPRO file corresponding to the Extended Sthocastic Symmetric Net (ESSN) (Pernice et al. 2019) exploited to model the COVID-19, and the solver (which should be generated before starting the analysis) are stored;
  3. cpp: the C++ code regarding the general transition of the ESSN;
  4. R_func: here are stored all the R scripts to generate the plots and to run the analysis;
  5. input: contains all the csv file (e.g., reference, parameter lists) necessary to run the analysis;
  6. Plot: contains the plots showed in the main paper.

COVID-19 model

Fig.1) SEIRS model and surveillance data on Piedmont region.

Fig.1) SEIRS model and surveillance data on Piedmont region.

In Figure 1 is showed:

  1. The age-dependent SEIRS model.
  2. The total infected cases distributed in the counties of the Piedmont region.
  3. Distribution of quarantine infected (Iq), hospitalized infected (Ih) and deaths (D) from February 24th to May 2nd. The control strategies are reported below the bar graph.

The population of the age class \(i\) is partitioned in the following seven compartments: (\(S_i\)), (\(E_i\)), (\(I_{ui}\)), (\(I_{qi}\)), (\(I_{hi}\)), (\(R_i\)), (\(D_i\)). With respect to the classical SEIRS model, we have added a transition from \(I_{ui}\) to \({I_{qi}}\) to model the possibility to identify undetected cases and isolate them. In this way an individual in \(I_{ui}\) tested as positive to the SARS-CoV-2 swab will be moved in the quarantine regime, \(I_{qi}\).

A detailed description of the model (e.g., system of ordinary differential equations, parameters, etc) is reported in the Supplementary Material.

Model Calibration

The calibration phase was performed to fit the model outcomes with the surveillance Piedmont infection and death data (from February 24st to May 2nd) using squared error estimator via trajectory matching. Hence, a global optimization algorithm, based on (Yang Xiang et al. 2012), was exploited to estimate 13 model parameters characterized by a high uncertainty due to their difficulty of being empirically measured:

  1. three parameters represent the probability of infection for each age class,
  2. four parameters reflect the governmental action strength at time epoch \(t\) (i.e., \(\alpha(t)\) with \(t\in \{\ \text{March 8th}, \text{March 21st}\}\)),
  3. one parameter describes the intensity of the population response (i.e., \(k\)),
  4. two parameters represent the death rate for the hospitalized patients (i.e., \(\sigma_i, \ i=2,\ 3\), fixing \(\sigma_1=0\)),
  5. two parameters are the initial condition for the undetected and quarantine infected individuals,
  6. the remainder parameter represents the detection rate for the third age class starting from the \(1^{st}\) April.

Consistently, Figure 2A and 2B show that the calibrated model is able to mimic consistently the observed infected and death cases (red line respectively). In Figure 3 the infected individuals for each age class are shown.

Fig.2)

Fig.2)

Fig.3)

Fig.3)

Studying the effects of the government control interventions.

Three scenarios are implemented. In the the model is calibrated to fit the surveillance data (yellow). In the the model extends the second restriction beyond March, \(21^{st}\) without implementing the third restriction (blue). In the the model consider a higher population compliance to the third governmental restriction (green).

Fig.4) Stochastic simulation results reported as traces (on the left) and as density distributions (on the right).

Fig.4) Stochastic simulation results reported as traces (on the left) and as density distributions (on the right).

COVID-19 epidemic containment strategies.

The daily evolution of infected individuals is shown varying on the columns the the efficacy of individual-level measures and on the rows the efficacy of community surveillance.

Fig.5)

Fig.5)

Fig.6)

Fig.6)

Specifically, in Figure 6 we show the daily forecasts of the number of infected individuals with the efficacy of individual-level measures ranging from \(0\%\) to \(60\%\) on the columns (increasing by steps of \(20\%\)) and, on the rows, increasing capability (from 0% to 30%, by 10% steps) of identifying otherwise undetected infected individuals. These results are obtained as median value of 5000 traces for each scenario obtained from the stochastic simulation.

References

Pernice, S., M. Pennisi, G. Romano, A. Maglione, S. Cutrupi, F. Pappalardo, G. Balbo, M. Beccuti, F. Cordero, and R. A. Calogero. 2019. “A Computational Approach Based on the Colored Petri Net Formalism for Studying Multiple Sclerosis.” BMC Bioinformatics.

Yang Xiang, Sylvain Gubian, Brian Suomela, and Julia Hoeng. 2012. “Generalized Simulated Annealing for Efficient Global Optimization: The GenSA Package for R.” The R Journal. http://journal.r-project.org/.